Why the Spectral Radius? An intuition-building introduction to the basic reproduction number

Bull Math Biol. 2022 Aug 5;84(9):96. doi: 10.1007/s11538-022-01057-9.


The basic reproduction number [Formula: see text] is a fundamental concept in mathematical epidemiology and infectious disease modeling. Loosely speaking, it describes the number of people that an infectious person is expected to infect. The basic reproduction number has profound implications for epidemic trajectories and disease control strategies. It is well known that the basic reproduction number can be calculated as the spectral radius of the next generation matrix, but why this is the case may not be intuitively obvious. Here, we walk through how the discrete, next generation process connects to the ordinary differential equation disease system of interest, linearized at the disease-free equilibrium. Then, we use linear algebra to develop a geometric explanation of why the spectral radius of the next generation matrix is an epidemic threshold. Finally, we work through a series of examples that help to build familiarity with the kinds of patterns that arise in parameter combinations produced by the next generation method. This article is intended to help new infectious disease modelers develop intuition for the form and interpretation of the basic reproduction number in their disease systems of interest.

Keywords: Basic reproduction number; infectious disease model; mathematical epidemiology; next generation matrix; spectral radius.

Publication types

  • Research Support, N.I.H., Extramural

MeSH terms

  • Basic Reproduction Number
  • Communicable Diseases* / epidemiology
  • Humans
  • Intuition*
  • Mathematical Concepts
  • Models, Biological
  • Radius